Hypersurfaces with constant mean curvature and prescribed area

Summary We consider—in the setting of geometric measure theory—hypersurfacesT (of codimension one) with prescribed boundaryB in Euclideann+1 space which maximize volume (i.e.T together with a fixed hypersurfaceT ₀ encloses oriented volume) subject to a mass constraint. We prove existence and optimal regularity of solutionsT of such variational problems and we show that, on the regular part of its support,T is a classical hypersurface of constant mean curvature. We also prove that the solutionsT become more and more spherical as the valuem of the mass constraint approaches ∞. This work was done at the Centre for Mathematics and its Applications at the Australian National University, Canberra while the author was a visiting member This article was processed by the author using the LAT_EX style filecljour1 from Springer-Verlag.